Theme Essential Question: How does mathematical reasoning apply to the attributes of two- and three- dimensional figures as seen in everyday life?
Essential Questions:
How can you solve problems using the formulas for volume?
Standards
8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
Objectives
The student will identify similarities between the volume of a rectangular prism with the volume of a cylinder, the relationship between the volume of a cone and a cylinder with same bases and height, and intuitively connect the volume of a cylinder with the volume of the sphere.
The students will know the formulas for the volumes of cones, cylinders, and spheres.
The student will solve problems using the formulas for volume.
Reflections and/or Comments from your PCSSD 8th Grade Curriculum Team
As the team reviewed the released Common Core documents, it began clear that students must have a level of experience making connections between the volumes of the various shapes. Help students develop the formulas instead of providing the formulas up front. This will make the concept more concrete instead of a plug and chug method.
In order to provide relevance to the lesson on square roots and cube roots, questions should be designed as an equation where the volume is given and students must solve to find the missing dimension.
For example, a distributer is trying to design a new cone container. Their research department has limited the height of the new cone container to 4 inches. If the container is to hold approximately 40 cubic inches, what would be the diameter of the container to the nearest tenth of an inch?
Background Information Recommended: For a quick overview of the standard(s) to be addressed in this lesson, see Arizona’s Content Standards Reference Materials. http://www.azed.gov/wp-content/uploads/PDF/MathGr8.pdf (Taken from Ohio Dept of Education Mathematics Model Curriculum 6-28-2011 ) Begin by recalling the formula, and its meaning, for the volume of a right rectangular prism: V = l ×w ×h. Then ask students to consider how this might be used to make a conjecture about the volume formula for a cylinder:
Most students can be readily led to the understanding that the volume of a right rectangular prism can be thought of as the area of a “base” times the height, and so because the area of the base of a cylinder is π r2 the volume of a cylinder is Vcylinder = π r2h. To motivate the formula for the volume of a cone, use cylinders and cones with the same base and height. Fill the cone with rice or water and pour into the cylinder. Students will discover/experience that 3 cones full are needed to fill the cylinder. This non-mathematical derivation of the formula for the volume of a cone, V = 1/3 π r2h, will help most students remember the formula.
For the volume of a sphere, it may help to have students visualize a hemisphere “inside” a cylinder with the same height and “base.” The radius of the circular base of the cylinder is also the radius of the sphere and the hemisphere. The area of the “base” of the cylinder and the area of the section created by the division of the sphere into a hemisphere is π r2. The height of the cylinder is also r so the volume of the cylinder is π r3. Students can see that the volume of the hemisphere is less than the volume of the cylinder and more than half the volume of the cylinder. Illustrating this with concrete materials and rice or water will help students see the relative difference in the volumes. At this point, students can reasonably accept that the volume of the hemisphere of radius r is 2/3 π r3 and therefore volume of a sphere with radius r is twice that or 4/3 π r3. There are several websites with explanations for students who wish to pursue the reasons in more detail. (Note that in the pictures above, the hemisphere and the cone together fill the cylinder.) Students should experience many types of real-world applications using these formulas. They should be expected to explain and justify their solutions.
Assessment Product
Students should complete the problem solving connections on page 137-140 in the Houghton Mifflin OnCore Mathematics Middle School Grade 8 book. The final presentation of the “Where in the park is Xander?” can be presented as a PowerPoint, poster, etc. This will conclude the end of this unit.
Key Questions
What are some strategies to recall the volumes of cones, cylinders, and spheres?
How are the volumes of cones, cylinders, and spheres applied to real-world problems?
Observable Student Behaviors
The student knows the formulas for the volumes of cones, cylinders, and spheres.
The student can use the formulas for the volumes of cones, cylinders, and spheres.
Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.
Vocabulary Volume
Suggested Activities [see Legend to highlight MCO and HYS]
Houghton Mifflin OnCore Mathematics Middle School Grade 8 Unit 5-7, p. 133-136
ABC Mastering the Common Core in Mathematics Chapter 13-1 through 13-6, p. 211-215
Gizmo Lessons
Investigating Angle Theorems-Activity A
Explore the properties of complementary, supplementary, vertical, and adjacent angles using a dynamic figure.
Polygon Angle Sum-Activity B
Derive the sum of the angles of a polygon by dividing the polygon into triangles and summing their angles. Vary the number of sides and determine how the sum of the angles changes. Dilate the polygon to see that the sum is unchanged. (Student Exploration Sheet & Teacher Guide Available)
Similar Polygons
Manipulate two similar figures and vary the scale factor to see what changes are possible under similarity.
Triangle Angle Sum-Activity A
Measure the angles of a triangle and find the sum. Then reshape and resize the triangle and confirm that the sum of angle measures is the same for all triangles.
Teaching the Common Core Math Standards with Hands-On Activities by Judith Muschla
8.G.9- p. 234
JBHM-8th GP1 (p. 326-342)
Glencoe Pre Algebra Volume: Pyramids and Cones 11-3 (p. 568-572)
Glencoe Geometry Volumes of Prisms and Cylinders 13-1 (p. 688-695)
Glencoe Geometry Volumes of Pyramids and Cones 13-2 (p. 696-701)
Glencoe Geometry Volume of Spheres 13-3 (p. 702-706)
The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.
Diverse Learners
Odyssey (teacher discretion)
Skills Tutor (teacher discretion)
Math’scool: Unit B Module 8
Homework
See appropriate Glencoe, Houghton Mifflin OnCore Mathematics Middle School Grade 8, JBHM, and ABC Mastering the Common Core in Mathematics materials
http://illuminations.nctm.org/LessonDetail.aspx?ID=L261 This Internet Mathematics Excursion is based on E-example 6.3.2 from the NCTM Principles and Standards for School Mathematics. This is the third in a sequence of four lessons designed for students to understand scale factor and volume of various rectangular prisms. In this lesson, the student can manipulate the scale factor that links two three-dimensional rectangular prisms and learn about the relationships between edge lengths and volumes
http://www.nctm.org/standards/content.aspx?id=26770 The user can manipulate the side lengths of one of two similar rectangles and the scale factor to learn about how the side lengths, perimeters, and areas of the two rectangles are related.
Side Length, Volume, and Surface Area of Similar Solids
http://www.nctm.org/standards/content.aspx?id=25097 The user can manipulate the scale factor that links two three-dimensional rectangular prisms and learn about the relationships among edge lengths, surface areas, and volumes
Linking Length, Perimeter, Area, and Volume
http://illuminations.nctm.org/LessonDetail.aspx?ID=U98 This cluster of Internet Mathematics Excursions is based on E-example 6.3 as described in the Geometry Standard and Measurement Standards. These lessons are designed for students to understand ratio, proportion, scale factor, and similarity using perimeter, area, volume and surface area of various rectangular shapes.
Scaling Away
http://illuminations.nctm.org/LessonDetail.aspx?ID=L584 Students will measure the dimensions of a common object, multiply each dimension by a scale factor, and examine a model using the multiplied dimensions. Students will then compare the surface area and volume of the original object and the enlarged model
http://illuminations.nctm.org/LessonDetail.aspx?ID=L639 Each student constructs a tetrahedron and describes the linear, area and volume using non‑traditional units of measure. Four tetrahedra are combined to form a similar tetrahedron whose linear dimensions are twice the original tetrahedron. The area and volume relationships between the first and second tetrahedra are explored, and generalizations for the relationships are developed.
Grade: 8 Unit: 2 Week: 8 Dates: 11/19-11/20 (2 days)
Content: Volume Formulas
Theme Essential Question:
How does mathematical reasoning apply to the attributes of two- and three- dimensional figures as seen in everyday life?
Essential Questions:
Standards
Objectives
Reflections and/or Comments from your PCSSD 8th Grade Curriculum Team
As the team reviewed the released Common Core documents, it began clear that students must have a level of experience making connections between the volumes of the various shapes. Help students develop the formulas instead of providing the formulas up front. This will make the concept more concrete instead of a plug and chug method.
In order to provide relevance to the lesson on square roots and cube roots, questions should be designed as an equation where the volume is given and students must solve to find the missing dimension.
For example, a distributer is trying to design a new cone container. Their research department has limited the height of the new cone container to 4 inches. If the container is to hold approximately 40 cubic inches, what would be the diameter of the container to the nearest tenth of an inch?
Background Information
Recommended: For a quick overview of the standard(s) to be addressed in this lesson, see Arizona’s Content Standards Reference Materials.
http://www.azed.gov/wp-content/uploads/PDF/MathGr8.pdf
(Taken from Ohio Dept of Education Mathematics Model Curriculum 6-28-2011 )
Begin by recalling the formula, and its meaning, for the volume of a right rectangular prism: V = l ×w ×h. Then ask students to consider how this might be used to make a conjecture about the volume formula for a cylinder:
Vcylinder = π r2h.
To motivate the formula for the volume of a cone, use cylinders and cones with the same base and height. Fill the cone with rice or water and pour into the cylinder. Students will discover/experience that 3 cones full are needed to fill the cylinder. This non-mathematical derivation of the formula for the volume of a cone, V = 1/3 π r2h, will help most students remember the formula.
For the volume of a sphere, it may help to have students visualize a hemisphere “inside” a cylinder with the same height and “base.” The radius of the circular base of the cylinder is also the radius of the sphere and the hemisphere. The area of the “base” of the cylinder and the area of the section created by the division of the sphere into a hemisphere is π r2. The height of the cylinder is also r so the volume of the cylinder is π r3. Students can see that the volume of the hemisphere is less than the volume of the cylinder and more than half the volume of the cylinder. Illustrating this with concrete materials and rice or water will help students see the relative difference in the volumes. At this point, students can reasonably accept that the volume of the hemisphere of radius r is 2/3 π r3 and therefore volume of a sphere with radius r is twice that or 4/3 π r3. There are several websites with explanations for students who wish to pursue the reasons in more detail. (Note that in the pictures above, the hemisphere and the cone together fill the cylinder.)
Students should experience many types of real-world applications using these formulas. They should be expected to explain and justify their solutions.
Assessment
Product
Key Questions
Observable Student Behaviors
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Vocabulary
Volume
Suggested Activities [see Legend to highlight MCO and HYS]
- Houghton Mifflin OnCore Mathematics Middle School Grade 8 Unit 5-7, p. 133-136
- ABC Mastering the Common Core in Mathematics Chapter 13-1 through 13-6, p. 211-215
Gizmo Lessons- Investigating Angle Theorems-Activity A
Explore the properties of complementary, supplementary, vertical, and adjacent angles using a dynamic figure.- Polygon Angle Sum-Activity B
Derive the sum of the angles of a polygon by dividing the polygon into triangles and summing their angles. Vary the number of sides and determine how the sum of the angles changes. Dilate the polygon to see that the sum is unchanged. (Student Exploration Sheet & Teacher Guide Available)- Similar Polygons
Manipulate two similar figures and vary the scale factor to see what changes are possible under similarity.- Triangle Angle Sum-Activity A
Measure the angles of a triangle and find the sum. Then reshape and resize the triangle and confirm that the sum of angle measures is the same for all triangles.- Teaching the Common Core Math Standards with Hands-On Activities by Judith Muschla
- 8.G.9- p. 234
- JBHM-8th GP1 (p. 326-342)
- Glencoe Pre Algebra Volume: Pyramids and Cones 11-3 (p. 568-572)
- Glencoe Geometry Volumes of Prisms and Cylinders 13-1 (p. 688-695)
- Glencoe Geometry Volumes of Pyramids and Cones 13-2 (p. 696-701)
- Glencoe Geometry Volume of Spheres 13-3 (p. 702-706)
Highly Recommended:- http://illustrativemathematics.org/illustrations/520
- http://illustrativemathematics.org/illustrations/517
- http://illustrativemathematics.org/illustrations/112
- http://illustrativemathematics.org/illustrations/521
The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.Diverse Learners
Homework
Terminology for Teachers
Ethnicity/Culture | Immigration/Migration | Intercultural Competence | Socialization | Racism/Discrimination
High Yield Strategies
Similarities/Differences | Summarizing/Notetaking | Reinforcing/Recognition | Homework/Practice |
Non-Linguistic representation | Cooperative Learning | Objectives/Feedback |
Generating-Testing Hypothesis | Cues, Questions, Organizers
Resources
Professional Texts
Literary Texts
Informational Texts
- See New York Common Core Aligned Task (other resources)
http://schools.nyc.gov/Academics/CommonCoreLibrary/SeeStudentWork/default.htmArt, Music, and Media
Manipulatives
Games
Videos
SMART Board Lessons, Promethean Lessons
Websites
Other Activities, etc.
- Fill'r Up
http://illuminations.nctm.org/LessonDetail.aspx?ID=L261This Internet Mathematics Excursion is based on E-example 6.3.2 from the NCTM Principles and Standards for School Mathematics. This is the third in a sequence of four lessons designed for students to understand scale factor and volume of various rectangular prisms. In this lesson, the student can manipulate the scale factor that links two three-dimensional rectangular prisms and learn about the relationships between edge lengths and volumes
- Finding Surface Area and Volume
http://illuminations.nctm.org/LessonDetail.aspx?ID=L609Using the isometric drawing tool, students build three-dimensional figures and find the surface area and volume of each figure
- Side Length and Area of Similar Figures
http://www.nctm.org/standards/content.aspx?id=26770The user can manipulate the side lengths of one of two similar rectangles and the scale factor to learn about how the side lengths, perimeters, and areas of the two rectangles are related.
- Side Length, Volume, and Surface Area of Similar Solids
http://www.nctm.org/standards/content.aspx?id=25097The user can manipulate the scale factor that links two three-dimensional rectangular prisms and learn about the relationships among edge lengths, surface areas, and volumes
- Linking Length, Perimeter, Area, and Volume
http://illuminations.nctm.org/LessonDetail.aspx?ID=U98This cluster of Internet Mathematics Excursions is based on E-example 6.3 as described in the Geometry Standard and Measurement Standards. These lessons are designed for students to understand ratio, proportion, scale factor, and similarity using perimeter, area, volume and surface area of various rectangular shapes.
- Scaling Away
http://illuminations.nctm.org/LessonDetail.aspx?ID=L584Students will measure the dimensions of a common object, multiply each dimension by a scale factor, and examine a model using the multiplied dimensions. Students will then compare the surface area and volume of the original object and the enlarged model
- Scaling Away
http://illuminations.nctm.org/Lessons/Scaling/Scaling-AS-ScalingAway.pdfThis reproducible worksheet, from an Illuminations lesson, contains questions regarding the effect of multiplying by a scale factor on the surface area and volume of a rectangular prism or cylinder
- Tetrahedral Kites
http://illuminations.nctm.org/LessonDetail.aspx?ID=L639Each student constructs a tetrahedron and describes the linear, area and volume using non‑traditional units of measure. Four tetrahedra are combined to form a similar tetrahedron whose linear dimensions are twice the original tetrahedron. The area and volume relationships between the first and second tetrahedra are explored, and generalizations for the relationships are developed.
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